Solidify non manifold (non contiguos)
The mesh object has quite a bit of double vertices shown in the example below. They will prevent modifier from working properly as if extruded they will create even more intersections leading to z-fighting even without solidifying.



Selecting all and removing doubles in Edit mode solves that as they are intersecting with each other. Then extrusion should be done to have faces Solidify to work with. After extruding by default Blender orients half of normals to one side and half - to another.



Blue lines show normals orientation. According to that modifier will orient half of faces to one side and half to another one leading to intersections. Select all and recalculate normals in Edit mode.

Lastly object scale is uneven. It doesn't prevent Solidify from working but it does give unexpected result, on the image below left is before scale applied, right one is after. To apply scale in Object mode press Ctrl+A.